【種花家務(wù)·代數(shù)】1-5-11本章復(fù)習(xí)（分式）『數(shù)理化自學(xué)叢書(shū)6677版』

2023-09-30 05:19 作者:山嵓 0人讀過(guò) | 我要投稿

【閱前提示】本篇出自『數(shù)理化自學(xué)叢書(shū)6677版』，此版叢書(shū)是“數(shù)理化自學(xué)叢書(shū)編委會(huì)”于1963-1966年陸續(xù)出版，并于1977年正式再版的基礎(chǔ)自學(xué)教材，本系列叢書(shū)共包含17本，層次大致相當(dāng)于如今的初高中水平，其最大特點(diǎn)就是可用于“自學(xué)”。當(dāng)然由于本書(shū)是大半個(gè)世紀(jì)前的教材，很多概念已經(jīng)與如今迥異，因此不建議零基礎(chǔ)學(xué)生直接拿來(lái)自學(xué)。不過(guò)這套叢書(shū)卻很適合像我這樣已接受過(guò)基礎(chǔ)教育但卻很不扎實(shí)的學(xué)酥重新自修以查漏補(bǔ)缺。另外，黑字是教材原文，彩字是我寫(xiě)的注解。

【山話嵓語(yǔ)】我在原有“自學(xué)叢書(shū)”系列17冊(cè)的基礎(chǔ)上又添加了1冊(cè)八五人教中學(xué)甲種本《微積分初步》，原因有二：一則，我是雙魚(yú)座，有一定程度的偶雙癥，但“自學(xué)叢書(shū)”系列中代數(shù)4冊(cè)、幾何5冊(cè)實(shí)在令我刺撓，因此就需要加入一本代數(shù)，使兩邊能夠?qū)ε计胶?；二則，我認(rèn)為《微積分初步》這本書(shū)對(duì)“準(zhǔn)大學(xué)生”很重要，以我的慘痛教訓(xùn)為例，大一高數(shù)第一堂課，我是直接蒙圈，學(xué)了個(gè)寂寞。另外大學(xué)物理的前置條件是必須有基礎(chǔ)微積分知識(shí)，因此我所讀院校的大學(xué)物理課是推遲開(kāi)課；而比較生猛的大學(xué)則是直接開(kāi)課，然后在緒論課中猛灌基礎(chǔ)高數(shù)（例如田光善舒幼生老師的力學(xué)課）。我選擇在“自學(xué)叢書(shū)”17本的基礎(chǔ)上添加這本《微積分初步》，就是希望小伙伴升大學(xué)前可以看看，不至于像我當(dāng)年那樣被高數(shù)打了個(gè)措手不及。

第五章分式??

本章提要

1、本章的重要概念

分式，最簡(jiǎn)分式（既約分式），約分，通分。

2、本章的重要法則

(1)分式的基本性質(zhì)： $%5Cscriptsize%5Cfrac%20ab%3D%5Cfrac%7Bam%7D%7Bbm%7D%3D%5Cfrac%7Ba%5Cdiv%20m%7D%7Bb%5Cdiv%20m%7D$ （m≠0）? 。

(2)分式的分子分母變換符號(hào)的法則： $%5Cscriptsize%5Cfrac%7B-a%7D%7B-b%7D%3D%5Cfrac%20ab%2C%5C%3B%5Cfrac%7B-a%7Db%3D%5Cfrac%20a%7B-b%7D%3D-%5Cfrac%20ab$ ? 。

(3)分式的加減法則

????①同分母分式的加減法： $%5Cscriptsize%5Cfrac%20ba%5Cpm%5Cfrac%20ca%3D%5Cfrac%7Bb%5Cpm%20c%7Da$ ? 。

????②分母不同的分式的加減法： $%5Cscriptsize%5Cfrac%20ba%5Cpm%5Cfrac%20dc%3D%5Cfrac%7Bbc%5Cpm%20ad%7D%7Bac%7D$ ? 。先通分，再按同分母分式加減法做。

(4)分式的乘法法則： $%5Cscriptsize%5Cfrac%7Ba%7D%7Bb%7D%5Ctimes%5Cfrac%7Bc%7D%7Bd%7D%3D%5Cfrac%7Bac%7D%7Bbd%7D$ ? 。

(5)分式的除法法則： $%5Cscriptsize%5Cfrac%7Ba%7D%7Bb%7D%5Cdiv%5Cfrac%7Bc%7D%7Bd%7D%3D%5Cfrac%7Ba%7D%7Bb%7D%5Ctimes%5Cfrac%7Bd%7D%7Bc%7D%3D%5Cfrac%7Bad%7D%7Bbc%7D$ ? 。

(6)繁分式的化簡(jiǎn)法則： $%5Cscriptsize%5Cfrac%7B%5Cfrac%20ab%7D%7B%5Cfrac%20cd%7D%3D%5Cfrac%20ab%5Ctimes%5Cfrac%20dc%3D%5Cfrac%7Bad%7D%7Bbc%7D$ ? 。

復(fù)習(xí)題五

1、怎樣的代數(shù)式叫做整式？怎樣的代數(shù)式叫做分式？寫(xiě)出兩個(gè)整式和兩個(gè)分式來(lái)。

2、整式的值一定是整數(shù)嗎？3x 的值可以是分?jǐn)?shù)嗎？舉一個(gè)例子。

3、分式的值一定是分?jǐn)?shù)嗎？ $%5Cscriptsize%5Cfrac%7B5%7D%7Bx%2B2%7D%20$ 的值可以是整數(shù)嗎？舉一個(gè)例子。

4、整式內(nèi)字母可以取任意值嗎？a2＋3a－5 內(nèi)的 a 可以取任意值嗎？

5、分式內(nèi)字母可以取任意值嗎？ $%5Cscriptsize%5Cfrac%7B5%7D%7Bx%2B2%7D%20$ 內(nèi)的 x 可以取任意值嗎？有什么限制？

6、敘述分式的基本性質(zhì)。

7、什么叫做最簡(jiǎn)分式（既約分式）？寫(xiě)出兩個(gè)最簡(jiǎn)分式來(lái)。寫(xiě)出兩個(gè)不是最簡(jiǎn)分式的分式來(lái)，再把這兩個(gè)分式化成最簡(jiǎn)分式。

8、下列約分方法是否正確？如果不正確，錯(cuò)在哪里？

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5Cfrac%7Bx%5E2%7D%7Bx%5E6%7D%3D%5Cfrac1%7Bx%5E3%7D%3B%26%26(2)%5Cfrac%7B4x%5E4%7D%7B8x%5E8%7D%3D%5Cfrac1%7B4x%5E4%7D%3B%5C%5C%26(3)%5Cfrac%7By%5E2%2Ba%5E2%7D%7Bx%5E2%2Ba%5E2%7D%3D%5Cfrac%7By%5E2%7D%7Bx%5E2%7D%3B%26%26(4)%5Cfrac%7By%5E2%7D%7Bx%5E2%7D%3D%5Cfrac%20yx.%5Cend%7Baligned%7D$

9、下列演算，是否正確？如果不正確，錯(cuò)在哪里？

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5Cfrac%7Bx%2B3%7Dx-%5Cfrac%7Ba-3%7Da%3D%5Cfrac%7Bax%2B3a-ax-3x%7D%7Bax%7D%3B%5C%5C%0A%26(2)%5Cfrac%20x%7Bx%2Ba%7D-%5Cfrac%20x%7Bx-a%7D%3D%5Cfrac%20x%7Bx%2Ba%7D%2B%5Cfrac%20x%7Bx%2Ba%7D.%0A%5Cend%7Baligned%7D$

10、下列演算，是否正確？如果不正確，錯(cuò)在哪里？

$%5Csmall(1)%5C%3B%5Cfrac%7B(a-x)%5E2%7D%7Bx%5E2-a%5E2%7D%3D%5Cfrac%7B-(x-a)%5E2%7D%7Bx%5E2-a%5E2%7D%3B%5C%3B(2)%5C%3B%5Cfrac%7Ba-x%7D%7B(a%5E2-x%5E2)%5E2%7D%3D%5Cfrac1%7B(a%2Bx)%5E2%7D.$

求下列代數(shù)式的值(11~22)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2611%E3%80%81%5Cfrac%7Ba%2Bb%7D%7Ba-b%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%20%7Da%3D-%5Cfrac12%2Cb%3D%5Cfrac13.%5C%5C%0A%2612%E3%80%81%5Cfrac%20ba-%5Cfrac%20ab%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%7Da%3D-3%2Cb%3D-5%5C%5C%0A%2613%E3%80%81%5Cfrac%7Bx%5E2%2B2y%5E2%7D%7B2x-3y%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%20%7Dx%3D0%2Cy%3D%5Cfrac13.%5C%5C%0A%2614%E3%80%81%5Cfrac%7Bx%5E2-2y%5E2%7D%7Bx-2y%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%20%7Dx%3D1%2Cy%3D-1.%5C%5C%0A%2615%E3%80%81%5Cfrac%7Bx%5E2-4y%5E2%7D%7Bx-4y%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%7Dx%3D-2%2Cy%3D-1.%5C%5C%0A%2616%E3%80%81%5Cfrac%7Bx%5E3-2x%5E2-3x%2B4%7D%7Bx%5E3%2B2x%5E2-3x%2B4%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%20%7Dx%3D1.%5C%5C%0A%2617%E3%80%81%5Cfrac%7Bx%5E3%2B2x%5E2-2x%2B5%7D%7Bx%5E3-2x%5E2%2B3x-1%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%20%7Dx%3D-%5Cfrac12.%5C%5C%0A%2618%E3%80%81%5Cfrac%7B(x%2B3)(x-5)%7D%7B(x%2B2)(x-1)%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%20%7Dx%3D-1%5Cfrac12.%5C%5C%0A%2619%E3%80%81%5Cfrac%7Bx%2B3%7D%7B%7Cx%7C%2B3%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%7Dx%3D-5.%5C%5C%0A%2620%E3%80%81%5Cfrac%7Bx-3%7D%7B-%7Cx%7C-3%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%7Dx%3D-3.%5C%5C%0A%2621%E3%80%81%5Cfrac%7Bx%2B3%7D%7B%7Cx-3%7C%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%7Dx%3D-1.%5C%5C%0A%2622%E3%80%81%5Cfrac%7B3x%7D%7B%7Cx-8%7C%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%7Dx%3D-5.%0A%5Cend%7Baligned%7D%0A$

在下列分式里，哪些字母的值有哪些限制(23~26)？

$%5Csmall23%E3%80%81%5Cfrac3x.%5C%3B24%E3%80%81%5Cfrac%7Bx%2B3%7D%7Bx-2%7D.%5C%3B25%E3%80%81%5Cfrac%7Bx-3%7D%7Bx%2B3%7D.%5C%3B26%E3%80%81%5Cfrac5%7B3-x%7D.$

化成既約分式(27~32)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2627%E3%80%81%5Cfrac%7Bx%5E5y%5E3-4x%5E3y%5E5%7D%7Bx%5E3y%5E2-2x%5E2y%5E3%7D.%5C%5C%0A%2628%E3%80%81%5Cfrac%7B(x%5E6-y%5E6)(x%2By)%7D%7B(x%5E3%2By%5E3)(x%5E4-y%5E4)%7D.%5C%5C%0A%2629%E3%80%81%5Cfrac%7Bx%5E2-4x-21%7D%7Bx%5E2%2B2x-63%7D.%5C%5C%0A%2630%E3%80%81%5Cfrac%7B3x%5E2-18bx%2B27%5Cboldsymbol%7Bb%7D%5E2%7D%7B2x%5E2-18b%5E2%7D.%5C%5C%0A%2631%E3%80%81%5Cfrac%7B(x%5E2-25)(x%5E2-8x%2B15)%7D%7B(x%5E2-9)(x%5E2-7x%2B10)%7D.%5C%5C%0A%2632%E3%80%81%5Cfrac%7B(x%5E2%2Bc%5E2)%5E2-4b%5E2x%5E2%7D%7Bx%5E4%2B4bx%5E3%2B4b%5E2x%5E4-c%5E4%7D.%0A%5Cend%7Baligned%7D%0A$

計(jì)算(33~42)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2633%E3%80%81%5Cfrac1%7B2a-3b%7D%2B%5Cfrac1%7B2a%2B3b%7D-%5Cfrac%7B6b%7D%7B4a%5E2-9b%5E2%7D.%20%5C%5C%0A%2634%E3%80%81%5Cfrac1%7Bx%2B1%7D%2B%5Cfrac1%7Bx%5E2-1%7D%2B%5Cfrac1%7Bx%5E3%2B1%7D.%20%5C%5C%0A%2635%E3%80%81%5Cfrac%7Bx%2B1%7D%7B(x-1)(x-2)%7D-%5Cfrac%7Bx%2B2%7D%7B(2-x)(x-3)%7D%2B%5Cfrac%7Bx%2B3%7D%7B(3-x)(1-x)%7D.%5C%5C%0A%2636%E3%80%81%5Cquad%5Cfrac%7Bx%5E3-1%7D%7Bx%5E3%2B1%7D%5Ccdot%5Cfrac%7Bx%2B1%7D%7Bx-1%7D.%5C%5C%0A%2637%E3%80%81%5Cquad%5Cleft(1-%5Cfrac%7Bx-2%7D%7Bx%5E2%2Bx-2%7D%5Cright)%5Ccdot%5Cfrac%7Bx%2B2%7Dx.%5C%5C%0A%2638%E3%80%81%5Cquad%5Cfrac%7Bx%5E2%2B5x%2B6%7D%7Bx%2B1%7D%5Cdiv(x%5E2%2B6x%2B8)%5C%5C%0A%2639%E3%80%81%5Cfrac%7B%5Cfrac%7Ba%2Bb%7D%7Ba-b%7D%2B1%7D%7B%5Cfrac%7Ba-b%7D%7Ba%2Bb%7D%2B1%7D.%5C%5C%0A%2640%E3%80%81%5Cfrac%7B%5Cfrac%20a%7Ba-b%7D-%5Cfrac%20b%7Ba%2Bb%7D%7D%7B%5Cfrac%20a%7Ba%2Bb%7D%2B%5Cfrac%20b%7Ba-b%7D%7D.%5C%5C%0A%2641%E3%80%81%5Cfrac%20a%7Bb%2B%5Cfrac%20c%7Bd%2B%5Cfrac%20ef%7D%7D.%5C%5C%0A%2642%E3%80%81%5Cfrac%7Bx%5E2-y%5E2-z%5E2%2B2yz%7D%7B%5Cfrac%7Bx-y%2Bz%7D%7Bx-y-z%7D%7D.%0A%0A%5Cend%7Baligned%7D$

約簡(jiǎn)下列分式，再求它的值(43~46)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2643%E3%80%81%5Cfrac%7B4a%5E2%2B8ab%2B4b%5E2%7D%7B2a%5E2-2b%5E2%7D%5Ctext%7B%2C%E5%85%B6%E4%B8%AD%7Da%3D6%5Cfrac7%7B40%7D%2Cb%3D-1.375.%20%5C%5C%0A%2644%E3%80%81%5Cfrac%7Bb%5E%7B3%7D-b%7D%7B(1%2Bab)%5E%7B2%7D-(a%2Bb)%5E%7B2%7D%7D%2C%5Ctext%7B%E5%85%B6%E4%B8%AD%7Da%3D-51%2Cb%3D26.%20%5C%5C%0A%2645%E3%80%81%5Cfrac%7Bm%5E2%2Bn%5E2-p%5E2%2B2mn%7D%7Bm%5E2-n%5E2%2Bp%5E2%2B2mp%7D%2C%5Ctext%7B%20%E5%85%B6%E4%B8%AD%20%7Dm%3D15%2Cn%3D-7%2Cp%3D-12.%20%5C%5C%0A%2646%E3%80%81%5Cfrac%7Bx%5E2-3x-18%7D%7Bx%5E2-12x%2B36%7D%5Ctext%7B%2C%E5%85%B6%E4%B8%AD%20%7Dx%3D23.%20%0A%0A%5Cend%7Baligned%7D$

化簡(jiǎn)(47~50)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2647.%5Cfrac3%7Bx%2B1%7D-%5Cfrac%7Bx-1%7D%7B%5Cfrac%7Bx%5E2%7D2%2B%5Cfrac%20x2-1%7D.%5C%5C%0A%2648.%5Cfrac%20x%7B1%2B%5Cfrac1x%7D%2B1-%5Cfrac1%7Bx%2B1%7D.%5C%5C%0A%2649.%5Cfrac%7B%5Cfrac1%7Bab%7D%2B%5Cfrac1%7Bac%7D%2B%5Cfrac1%7Bbc%7D%7D%7B%5Cfrac%7Ba%5E2-(b%2Bc)%5E2%7D%7Bab%7D%7D.%5C%5C%0A%2650.%5Cfrac3%7B1%2B%5Cfrac3%7B1%2B%5Cfrac3%7B1-x%7D%7D%7D.%0A%0A%5Cend%7Baligned%7D$

*化簡(jiǎn)(51~52)：

$%5Csmall%5Cbegin%7Baligned%7D%2651%E3%80%81%5Cleft(%5Cfrac%7Bx-y%7D%7B2y-x%7D-%5Cfrac%7Bx%5E2%2By%5E2%2By-2%7D%7Bx%5E2-xy-2y%5E2%7D%5Cright)%5Cdiv%5Cfrac%7B4x%5E4%2B4x%5E2y%2By%5E2-4%7D%7Bx%5E2%2By%2Bxy%2Bx%7D.%5C%5C%2652%E3%80%81%5Cfrac%7Ba%5E2%2Ba-2%7D%7Ba%5E%7Bn%2B1%7D-3a%5En%7D%5Ccdot%5Cleft%5B%5Cfrac%7B(a%2B2)%5E2-a%5E2%7D%7B4a%5E2-4%7D-%5Cfrac3%7Ba%5E2-a%7D%5Cright%5D.%5Cend%7Baligned%7D$

【答案】

$%5Cscriptsize%5Cbegin%7Baligned%7D%0A%2611%E3%80%81%5Cfrac%7B1%7D%7B5%7D%3B%5C%3B12%E3%80%811%5Cfrac%7B1%7D%7B15%7D%3B%5C%3B13%E3%80%81-%5Cfrac%7B2%7D%7B9%7D%3B%5C%3B14%E3%80%81-%5Cfrac%7B1%7D%7B3%7D%3B%5C%3B15%E3%80%810%3B%5C%3B16%E3%80%810%3B%5C%5C%2617%E3%80%81-2%5Cfrac%7B1%7D%7B25%7D%3B%5C%3B18%E3%80%817%5Cfrac%7B4%7D%7B5%7D%3B%5C%3B19%E3%80%81-%5Cfrac%7B1%7D%7B4%7D%3B%5C%3B20%E3%80%811%3B%5C%3B21%E3%80%81%5Cfrac%7B1%7D%7B2%7D%3B%5C%5C%2622%E3%80%81-1%5Cfrac%7B2%7D%7B13%7D%3B%5C%3B23%E3%80%81%20x%5Cneq0%3B%5C%3B24%E3%80%81%20x%5Cneq2%3B%5C%3B25%E3%80%81%20x%5Cneq-3%3B%5C%3B26%E3%80%81%20x%5Cneq3%3B%5C%5C%0A%2627%E3%80%81%20xy(x%2B2y)%3B%5C%3B28%E3%80%81%5Cfrac%7Bx%5E%7B3%7D%2Bxy%2By%5E%7B3%7D%7D%7Bx%5E%7B2%7D%2By%5E%7B2%7D%7D%3B%5C%3B29%E3%80%81%5Cfrac%7Bx%2B3%7D%7Bx%2B9%7D%3B%5C%3B30%E3%80%81%5Cfrac%7B3(x-3b)%7D%7B2(x%2B3b)%7D%3B%20%20%5C%5C%0A%2631%E3%80%81%5Cfrac%7Bx%5E%7B2%7D-25%7D%7B(x%2B3)(x-2)%7D%3B%5C%3B32%E3%80%81%5Cfrac%7Bx%5E%7B2%7D-2bx%2Bc%5E%7B2%7D%7D%7Bx%5E%7B2%7D%2B2bx-c%5E%7B2%7D%7D%3B%5C%3B33%E3%80%81%5Cfrac%7B2%7D%7B2a%2B3b%7D%3B%20%20%5C%5C%0A%2634%E3%80%81%5Cfrac%7Bx%5E3-x%5E2%2B2x-1%7D%7B(x%2B1)(x-1)(x%5E3-x%2B1)%7D%3B%5C%3B35%E3%80%81%5Cfrac%7B3x%5E2-11%7D%7B(x-1)(x-2)(x-3)%7D%3B%20%20%5C%5C%0A%2636%E3%80%81%5Cfrac%7Bx%5E2%2Bx%2B1%7D%7Bx%5E2-x%2B1%7D%3B%5C%3B37%E3%80%81%5Cfrac%20x%7Bx-1%7D%3B%5C%3B38%E3%80%81%5Cfrac%7Bx%2B3%7D%7B(x%2B1)(x%2B4)%7D%3B%5C%3B39%E3%80%81%5Cfrac%7Ba%2Bb%7D%7Ba-b%7D%3B%20%5C%5C%0A%2640%E3%80%811%3B%5C%3B41%E3%80%81%5Cfrac%7Badf%2Bae%7D%7Bbdf%2Bbe%2Bcf%7D%3B%5C%3B42%E3%80%81(x%2By-z)(x-y-z)%3B%5C%3B%2043%E3%80%811%5Cfrac%7B41%7D%7B151%7D%3B%5C%5C%0A%2644%E3%80%81%5Cfrac%7B1%7D%7B100%7D%3B%5C%3B45%E3%80%812%3B%5C%3B46%E3%80%811%5Cfrac%7B9%7D%7B17%7D%3B%5C%3B47%E3%80%81%5Cfrac%7Bx%2B4%7D%7B(x%2B1)(x%2B2)%7D%3B%20%5C%3B48%E3%80%81%20x%3B%5C%5C%0A%2649%E3%80%81%5Cfrac1%7Bc(a-b-c)%7D%3B%5C%3B50%E3%80%81%5Cfrac%7B12-3x%7D%7B7-4x%7D%3B%5C%3B%2051%E3%80%81%5Cfrac%7Bx%2B1%7D%7B(2y-x)(2x%5E2%2By%2B2)%7D%3B%5C%3B52%E3%80%81%5Cfrac%7Ba%2B2%7D%7Ba%5E%7Bn%2B1%7D%7D.%0A%5Cend%7Baligned%7D$

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【種花家務(wù)·代數(shù)】1-5-11本章復(fù)習(xí)（分式）『數(shù)理化自學(xué)叢書(shū)6677版』

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復(fù)習(xí)題五